Worksheet: Points, Midpoints, and Distances in Space

In this worksheet, we will practice finding the coordinates of a point in 3D, the distance between two points in 3D, and the coordinates of a midpoint and an endpoint in 3D using the formula.

Q1:

Given that the midpoint of 𝐴𝐵 lies in the 𝑥𝑦-plane, and the coordinates of 𝐴 and 𝐵
are (−12,−9,𝑘+3) and (−15,−9,3𝑘), respectively, determine the value of 𝑘.

A−34

B34

C−43

D43

Q2:

Points 𝐴, 𝐵 have coordinates (8,−8,−12) and (−8,5,−8), respectively. Determine the coordinates of the midpoint of 𝐴𝐵.

A0,−32,−10

B(0,−3,−20)

C(8,8,−2)

D(16,−13,−4)

Q3:

Determine, to the nearest hundredth,
the perimeter of the triangle formed by joining the midpoints of the sides of
△𝐴𝐵𝐶, given that
the coordinates of 𝐴, 𝐵, and 𝐶 are
(−10,−8,2),
(−8,−7,10),
and (−2,3,−14), respectively.

Q4:

Given that point (0,17,−10) is the midpoint of 𝐴𝐵 and that 𝐴(−19,7,14), what are the coordinates of 𝐵?

A(19,27,−34)

B(−19,24,4)

C(−19,41,−6)

D(19,10,−24)

Q5:

Given that point (−9,17,11) is the midpoint of 𝐴𝐵 and that 𝐴(4,−2,9), what are the coordinates of 𝐵?

A(−22,36,13)

B(−5,15,20)

C(−14,32,31)

D(−13,19,2)

Q6:

Given that point (−1,4,−18) is the midpoint of 𝐴𝐵 and that 𝐴(12,8,−1), what are the coordinates of 𝐵?

A(−14,0,−35)

B(11,12,−19)

C(10,16,−37)

D(−13,−4,−17)

Q7:

Determine, to the nearest hundredth,
the perimeter of the triangle formed by joining the midpoints of the sides of
△𝐴𝐵𝐶, given that
the coordinates of 𝐴, 𝐵, and 𝐶 are
(19,−18,4),
(1,−4,−16),
and (13,18,−3), respectively.

Q8:

Given that the midpoint of 𝐴𝐵 lies in the 𝑥𝑧-plane, and the coordinates of 𝐴 and 𝐵
are (−14,𝑘+4,−19) and (17,2𝑘,18), respectively, determine the value of 𝑘.

A−43

B43

C−34

D34

Q9:

Given that the midpoint of 𝐴𝐵 lies in the 𝑥𝑦-plane, and the coordinates of 𝐴 and 𝐵
are (3,−18,𝑘+5) and (19,1,5𝑘), respectively, determine the value of 𝑘.

A−56

B56

C−65

D65

Q10:

In which of the following coordinate planes does the point (−7,−8,0) lie?

A𝑦𝑧

B𝑥𝑦

C𝑥𝑧

Q11:

Given that the point (𝑥,𝑦,𝑧) lies in the 𝑥𝑦-plane, determine its 𝑧-coordinate.

Q12:

Given that the point (𝑥,𝑦,𝑧) lies in the 𝑦𝑧-plane, determine its 𝑥-coordinate.

Q13:

Given that the point (𝑥,𝑦,𝑧) lies in the 𝑥𝑧-plane, determine its 𝑦-coordinate.

Q14:

Determine the coordinates of point 𝐴.

A𝐴(−3,3,3)

B𝐴(3,−3,4)

C𝐴(3,−3,3)

D𝐴(0,0,4)

Q15:

Given that the point (7−𝑚,12−𝑚,𝑚) lies in the
𝑦𝑧-plane, what is this point?

A(0,12,7)

B(0,7,7)

C(0,5,7)

D(7,12,0)

Q16:

In the figure shown, the points 𝑂 and 𝐴 have coordinates (0,0,0) and (7,5,6), respectively. Determine the coordinates of 𝐵 and 𝐶.

A𝐵(0,7,5), 𝐶(0,7,6)

B𝐵(0,0,6), 𝐶(0,5,0)

C𝐵(7,0,5), 𝐶(7,5,0)

D𝐵(7,5,0), 𝐶(7,0,6)

Q17:

What is the distance between the point (19,5,5) and the 𝑥-axis?

A19 length units

B√411 length units

C5√2 length units

D√10 length units

Q18:

Calculate, to two decimal places, the area of the triangle 𝑃𝑄𝑅,
where the coordinates of its vertices are at 𝑃(4,0,2), 𝑄(2,1,5),
and 𝑅(−1,0,1).

Q19:

The points 𝐴, 𝐵, and 𝐶 are
on the 𝑥-, 𝑦-, and 𝑧-axes, respectively. Given that (12,−12,0) is the
midpoint of 𝐴𝐵 and (0,−12,−14) the midpoint of 𝐵𝐶, find the coordinates of the midpoint of 𝐴𝐶.

A(6,0,7)

B(12,0,−14)

C(24,0,−28)

D(6,−12,−7)

Q20:

Given that 𝐶−12,0,−2 is the midpoint of
𝐴𝐵, where the coordinates of 𝐴 and
𝐵 are (𝑘+5,8,𝑚+4) and
(−6,𝑛+7,5), respectively, what is
𝑘+𝑚−𝑛?

Q21:

Given that point (5𝑎,𝑎+2,−14) lies in the 𝑥𝑧-plane, determine its distance from the 𝑦𝑧-plane.

Q22:

Given that 𝐴(𝑎,𝑏,𝑐) is the midpoint of the line segment between 𝐵(9,−17,2) and 𝐶(16,−12,7), what is 𝑎+𝑏+𝑐?

A32

B452

C−172

D52

Q23:

Find the distance between the two points 𝐴(−7,12,3) and 𝐵(−4,−1,−8).

A√267length units

B√299length units

C299length units

D267length units

Q24:

Find 𝑘 so that the points (3,9,−4), (9,−3,−1), (−7,29,𝑘) are collinear.

Q25:

Given that the points (6,4,3), (7,6,𝑘), and (5,5,1)
are the vertices of a triangle, determine all the possible values of 𝑘 that make the triangle equilateral.